I am using Office 365 (Sharepoint Online) and I am using there also Infopath Form Web Parts
Can I use a Published Infopath form as the basis of a webpart
For example, in browser (or Sharepoint Designer 2010) create a web page; on ribbon, Site Actions > New Page > enter "New page name:" > click Create button > on ribbon under Insert tab press More Web Parts > Forms > Infopath Form Web Part > Add...
In Infopath Form Web Part Menu choose Edit Web Part > Select a list or a library... > etc.
Can this form be made as a webpart or do I have to duplicate all fields as a list and use the list form as the basis of the webpart
This depends ... or I didn't understand clearly what you've asked
What I find is that when I try to create a webpart in Office 365 I cannot find the library
Infopath form web part is usually used in web pages and in whatever locations you have chosen not in the library
Then, while creating an Infopath Form Web Part in web page, you just insert/link/cpnfigure previously created in Infopath Designer 2010 Infopath form (in other words, one cannot change/create/modify Infopath form outside of Infopath Designer 2010)
This may be due to the fact that there are allot of sub sites for different companie sites all on the same server as part of the cloud
One has no access to content of other subscribers in Office 365.
You probably did not get familiar with your own content in your Office 365 site collection generated for you automatically
The forms publish to the subsite and not the main root directory?
Anywhere you want, but location of web page to which you inserted Infopath Form Web Part has even more loose location of your own choice